CN107645335A

CN107645335A - MPPM FSO systems based on EW composite channel models demodulate error sign ratio computational methods firmly

Info

Publication number: CN107645335A
Application number: CN201710814265.9A
Authority: CN
Inventors: 秦亦灵; 王平; 王坤; 刘晓霞; 傅慧华
Original assignee: Xidian University
Current assignee: Xidian University
Priority date: 2017-09-11
Filing date: 2017-09-11
Publication date: 2018-01-30
Anticipated expiration: 2037-09-11
Also published as: CN107645335B

Abstract

The invention discloses a kind of MPPM FSO systems based on EW composite channel models to demodulate error sign ratio computational methods firmly, including：1) it is distributed based on EW, it is contemplated that the influence of point tolerance, establish point-to-point FSO composite channel model；2) threshold value of optimal judgement thresholding and dynamic decision thresholding is determined according to the cumulative distribution model and probability Distribution Model of EW turbulent flows and point tolerance composite channel model；3) probability of mistaken verdict of each time slot under by EW turbulent flows and point tolerance compound channel disturbed condition is calculated；4) Gauss Laguerre polynomials are utilized, obtain the closure expression formula of error sign ratio.Composite channel model in the inventive method considers influence of the point tolerance to systematic function compared to single atmospheric turbulance model, and this method complexity is low, is better than other hard demodulating algorithms in error sign ratio systematic function, more tallies with the actual situation.

Description

MPPM-FSO systems based on EW composite channel models demodulate error sign ratio calculating firmly Method

Technical field

Index Weibull distribution, which is based on, the present invention relates to (N, M) the MPPM modulation systems in radio optical communication system is considering point Computational methods are demodulated using optimal judgement thresholding and dynamic decision thresholding firmly in the case of error influence, belong to wireless light communication Technical field.

Background technology

Wireless light communication, and often referred to as FSO (Free space optical Communication), i.e. FSO communicates.FSO communications are using light wave as carrier, and communication is transmitted in free space transmission medium A kind of new broadband wireless communications means of information, it has, and cost is cheap, sets up flexibly convenient, anti-electromagnetic interference capability By force, good confidentiality, without frequency resource license, bandwidth the advantages that, receive everybody extensive concern.Although wireless light communication There is its unique advantage, but it also receives the influence and restriction of factors.In FSO systems, light beam using air as Transmission medium, and atmospheric environment is complicated and changeable, communication link can undoubtedly be disturbed by atmosphere.Up to the present, great Liang Yan Study carefully and show that atmospheric turbulance is one of main factor for influenceing communication performance, the effect can cause signal that amplitude occurs in receiving terminal With the randomized jitter of phase, and then cause obvious light intensity scintillation effect, greatly reduce communication system performance.FSO communication systems System also receives the restriction of other non-atmospheric factors except this principal element is influenceed by atmospheric turbulance.Point tolerance is another The principal element of one restriction communication quality, it is caused by the randomized jitter factor such as change of thermal expansion, slight earthquake, wind speed Transmitting terminal and caused by receiving terminal fixed platform rocks at random.And light beam is all narrower used by radio optical communication system, Random rock that hair receives both ends platform can cause the non-aligned of light beam, so as to get and random decay occurs for the optical signal up to receiving terminal, And then it have impact on communication link performance.In order to preferably study the influence that air declines to FSO systems, scientific research personnel successively proposes Different atmospheric channel models, such as logarithm normal distribution (log-normal, LN) model (being applied to weak rapid situation), K points Cloth model (be suitable to strong rapid condition), and double gammas (Gamma-Gamma, GG) distributed model (be applied to by force, weak three rapid feelings Condition).Wherein, GG models are of great interest and study, and the mathematical form of GG distributed models is easily handled, and is being used During the receiver of small-bore, notional result can coincide with the emulation under three kinds of turbulent flow conditions of weak, medium, strong with experimental data well. But when considering aperture averaging effect, in GG distributed models in the case of strong rapid result and experimental data it is identical be not Very well.2012, R.Barrios and F.Dios were based on Weibull distribution model, the relevant nature transmitted with reference to laser in turbulent flow With the physical significance of aperture averaging, it is proposed that index weber (exponentiated Weibull, EW) model.By and it is different Pore size is contrasted from the emulation under different turbulence intensities and experimental data, show the model can any aperture and appoint The random fluctuation situation of light intensity in receiver plane is described in the case of turbulent flow of anticipating.

Current digital optical communication system be mostly intensity modulated/directly detection (IM/DD) system, conventional modulation system There are on-off keying (on-off keying, OOK) and pulse position modulation (pulse position modulation, PPM).And New multi-pulse position modulation (Multipulse Pulse Position Modulation, MPPM) mode, relative to OOK With bandwidth availability ratio and power efficiency can be improved for PPM.

Hard demodulation is a kind of conventional demodulation method in free-space optical communication system, and it is made an uproar by decision threshold handle Sound and the signal determining of influence of fading are corresponding modulation level.And fixed sentence thresholding is wherein most simple and easy to get, but hold It is also easy to produce error thresholds.And optimal judgement thresholding and dynamic decision thresholding can eliminate this error thresholds, the standard of judgement is improved Exactness, so as to lifting system performance.

At present the problem of, is：Due to MPPM complexity, the report on its hard demodulation characteristics is less.It is currently known hard Demodulating system algorithm is the effect for only considering atmospheric turbulance mostly, without the influence of consideration point tolerance.It is thus impossible to for referring to Lead actual FSO communication system receivers design.It is contemplated that to the compound action of point tolerance and EW distribution atmospheric turbulances, it is based on MPPM modulation carries out hard demodulation judgement using optimal judgement thresholding and dynamic decision thresholding, calculates the erratum of radio optical communication system Number rate performance current highly important research direction of category in fact.

The content of the invention

The compound action of consideration point tolerance and EW distribution atmospheric turbulances, using optimal judgement thresholding and is moved based on MPPM modulation State decision threshold carries out hard demodulation judgement, calculates the error sign ratio performance of radio optical communication system category is current in fact and highly important grind Study carefully direction.Wherein, optimal judgement thresholding and the hard demodulating algorithm of dynamic decision thresholding can eliminate error thresholds, and raising system is sentenced Accuracy certainly.A kind of mathematical modeling of the error sign ratio performance estimation method proposed, the hard demodulation to above-mentioned (N, M) MPPM are calculated Method error sign ratio carries out theory analysis, and then correctly effectively calculates theoretical error sign ratio.

The present invention is realized by following technical proposals.

MPPM-FSO systems of the invention based on EW composite channel models demodulate error sign ratio computational methods firmly, including following Step：

1) it is distributed based on index weber EW, it is contemplated that the influence of point tolerance, establish point-to-point FSO and answer Close channel model；

2) according to index weber EW turbulent flows and the cumulative distribution model F of the composite model of point tolerance_hAnd probability distribution mould (h) Type f_h(h) threshold value of optimal judgement thresholding and dynamic decision thresholding is determined；

3) mistake of each time slot under by index weber EW turbulent flows and point tolerance compound channel disturbed condition is calculated to sentence Probability certainly；

4) Gauss Laguerre polynomials are utilized, obtain the closure expression formula of error sign ratio.

Further, in step 1), based on index weber EW be distributed, it is contemplated that the influence of point tolerance, establish it is point-to-point from By space optical communication composite channel model：If for x to send signal sequence, y is reception signal sequence, then obtains the mathematical modulo of channel Type, the probability Distribution Model of index Weibull distribution model is built according to obtained mathematical modelingWith the probability point of point tolerance Cloth modelIt then can obtain the cumulative distribution model F of the composite channel model of atmospheric turbulance and point tolerance_h(h) it is and general Rate distributed model f_h(h)。

Further, in the step 2), the threshold settings of optimal judgement thresholding utilize Matlab by dichotomy under Formula seeks unknown number y_thObtain.

Further, the threshold value of dynamic decision thresholding is to detect the signal of each time slot, takes its signal leading edge 200ns, the value of setting dynamic decision thresholding is taken according to the signal intensity detected in this period.

Further, in the step 3), each time slot is by index weber EW turbulent flows and the interference of point tolerance compound channel In the case of mistaken verdict, realized by following methods：

When 3a) using optimal judgement thresholding, it is known that the probability Distribution Model f of compound channel_h(h) numeral and in step 1) Model, then can calculate non-slot signal f (y | y₀) and time slot signal f (y | y₁) sample value probability density function, and then obtain Non-slot signal f (y | y₀) mistaken verdict probability, i.e. false-alarm probability P_fWith time slot signal f (y | y₁) mistaken verdict probability, i.e., False dismissal probability P_m；

When 3b) using dynamic decision thresholding, it is known that the probability that (N, M) MPPM block of informations empty slots occur is N-M/N, non- The probability that empty slot occurs is M/N, then probability of miscarriage of justice of each time slot when existing for only white Gaussian noise；Known compound letter Mathematical modeling in the probability Distribution Model and step 1) in road, then each time slot can be calculated under by compound channel disturbed condition Mistaken verdict probability.

Further, in the step 4), the closure expression formula for obtaining error sign ratio specifically includes：

When 4a) using optimal judgement thresholding, for (N, M) MPPM, according to the probability P correctly demodulated_c=(1-P_m)^M(1- P_f)^N-M, obtain error sign ratio；

When 4b) using dynamic decision thresholding, for (N, M) MPPM, the probability correctly demodulated is P_c=(1-SLER)^N, obtain Error sign ratio.

The present invention has advantages below：

A kind of MPPM-FSO systems based on EW composite channel models are first proposed in the present invention and demodulate error sign ratio firmly Computational methods, this method complexity is low, is better than other hard demodulating algorithms in error sign ratio systematic function；

Composite channel model in the inventive method considers point tolerance to system compared to single atmospheric turbulance model The influence of performance, more tallies with the actual situation.

Brief description of the drawings

Fig. 1 is MPPM modulation demodulation systems model under compound channel；

Fig. 2 is consideration point tolerance and the error sign ratio comparison diagram for not considering point tolerance in the case of strong rapid and weak two kinds of turbulent flow；

Fig. 3 is error sign ratio theoretical value comparison diagram when aperture diameter changes in the case of middle rapids；

Fig. 4 is error sign ratio theoretical value comparison diagram when shake variance changes under weak turbulent flow condition；

Fig. 5 is that optimal judgement threshold schemes, mistake when beam angle changes are used in the case of strong rapid and weak two kinds of turbulent flow Symbol rate theoretical value comparison diagram；

Fig. 6 is that dynamic decision door scheme, erratum when beam angle changes are used in the case of strong rapid and weak two kinds of turbulent flow Number rate theory value comparison diagram.

Embodiment

To make the object, technical solutions and advantages of the present invention clearer, enter with reference to the accompanying drawings and detailed description One step describes in detail.The present embodiment only represents the schematic illustration to the present invention, does not represent any limitation of the invention.

Hard demodulation bit error rate computational methods of the present invention based on MPPM modulation using two kinds of decision thresholds, including：

Step 1 is distributed based on index weber EW, it is contemplated that the influence of point tolerance, establishes point-to-point FSO Composite channel model

X 1a) is set to send signal sequence, y is reception signal sequence, then the mathematical modeling of channel is：

Y=Rhx+n

Wherein, n represent average be 0, variance beAdditive white Gaussian noise.R is photodetector responsiveness, and h is clothes From the channel fading of index weber EW compound channels distribution, h=h_ah_p；Wherein, h_aTo obey the channel fading of EW distributions, h_pFor Point tolerance loss factor.

1b) according to step 1a) probability Distribution Models of obtained mathematical modeling structure index weber EW distributed modelsWith the probability Distribution Model of point toleranceIt is expressed as

Wherein, h_aTo obey the channel fading of index weber EW distributions, h_pFor point tolerance loss factor；β is form parameter, β ＞ 0, its value are related to scintillation index；α is in given observation space, collimates the light beam of propagation and the light beam quilt of non-aligned propagation The average magnitude being properly received, α ＞ 0；η scale parameters, η ＞ 0；γ be equivalent beam angle and shake standard deviation ratio, γ= w_zeq/2σ_s；Wherein w_zeqFor equivalent beam angle, σ_sTo shake standard deviation；A₀For when the skew between detector and beam center Measure for 0 when the light intensity that receives of receiver, A₀=[erf (v)]², wherein ν is the ratio of the beam angle at aperture and distance z,Wherein a be detector aperture radius, w_zFor the beam angle at the z of range transmission end.

The cumulative distribution model F of index weber EW compound channels 1c) can be then derived according to the two formulas_h(h) it is and general Rate distributed model f_h(h) it is：

In formula, Γ (a, x) is upper gamma incomplete function,For Mayer G-function；J is the meter of summation symbol Number device.

Step 2 is according to index weber EW turbulent flows and the cumulative distribution model F of the composite channel model of point tolerance_h(h) it is and general Rate distributed model f_h(h) threshold value of optimal judgement thresholding and dynamic decision thresholding is determined

2a) threshold value of optimal judgement thresholding can seek unknown number y by dichotomy using Matlab to following formula_thObtain

In formula, number of time slots that a block of information that N is modulated by MPPM includes, M is the pulse in this N number of number of time slots Number；T is the integration variable of integrand；y_thFor threshold value；μ is coefficient, and μ²=(RNP_t/M)², R is photoelectric transformation efficiency, P_tFor the average emitted power of transmitting terminal；G is signal to noise ratio；Exp () is the exponential function using natural number e the bottom of as；Γ () is Gamma function；Erf () is error function；Erfc () is cosine error function；J is the counter of summation symbol.

2b) dynamic decision thresholding is the signal for detecting each time slot, the 200ns of its signal leading edge is taken, according to this period The signal intensity inside detected takes the value of setting dynamic decision thresholding.By the detection being worth to forward position, then make decisions thresholding Setting has been equivalently employed without the influence of turbulent flow, influences the only white Gaussian noise of conclusive judgement.

Step 3 calculates mistake of each time slot under by index weber EW turbulent flows and point tolerance compound channel disturbed condition The probability of judgement

When 3a) using optimal judgement thresholding, it is known that the probability Distribution Model f of compound channel_h(h) mathematics and in step 1) Model, then can calculate by go out under index weber EW turbulent flows and point tolerance disturbed condition non-slot signal f (y | y₀) believe with time slot Number f (y | y₁) probability density function of sample value is respectively：

In formula, σ_n ²For the variance of additive white Gaussian noise；f(y|y₁) it is the probability density letter for having signal slot sample value Number, and f (y | y₀) be non-signal time slot sample value probability density function；

False-alarm probability P_fThe probability that non-slot signal error is adjudicated, it is：

False dismissal probability P_mI.e. by the probability of time slot signal mistaken verdict, it is：

When 3b) using dynamic decision thresholding, it is known that the probability that (N, M) MPPM block of informations empty slots occur is N-M/N, non- The probability that empty slot occurs is M/N, then probability of miscarriage of justice of each time slot when existing for only white Gaussian noise is

The probability Distribution Model f of known compound channel_h(h) mathematical model and in step 1), then each time slot can be calculated The probability of mistaken verdict under by compound channel disturbed condition,

Step 4 utilizes Gauss Laguerre polynomials, obtains the closure expression formula of error sign ratio

When 4a) using optimal judgement thresholding, for (N, M) MPPM, the probability correctly demodulated is P_c=(1_-P_m)^M(1_-P_f )^N-M, wherein pulse slot and the correct probability of non-pulse time slot be

Then error sign ratio is

In formula, H_iIt is the weight in generalized Laguerre formula；x_iFor i-th of Generalized Gaussian Laguerre polynomials；K is The total item of Generalized Gaussian Laguerre polynomials.

When 4b) using dynamic decision thresholding, for (N, M) MPPM, the probability correctly demodulated is P_c=(1-SLER)^N, then miss Symbol rate is

The correctness and advantage of the present invention can be contrasted by following notional result and further illustrated：

Analytical Calculation, first, the formula that accurate description is derived are carried out by MATLAB；Then, compare have point tolerance and There is no the situation of point tolerance；Secondly, the performance of optimal judgement thresholding and dynamic decision thresholding is compared；Finally, change wherein various The value of variable.

Notional result

Fig. 1 is given under the compound channel of index weber EW turbulent flows and point tolerance, and that is modulated and demodulated using MPPM is wireless Optical communication system model.Fig. 2 uses optimal judgement thresholding and dynamic decision threshold schemes under the conditions of being given at strong rapid and weak rapids, examines Consider point tolerance and do not consider the comparison diagram that the error sign ratio in the case of two kinds of point tolerance changes with signal to noise ratio, it can be seen that consideration Point tolerance is decreased obviously than bit error rate performance in the case of not considering point tolerance.Fig. 3 uses optimal judgement in the case of being given at middle rapids Thresholding and dynamic decision threshold schemes, the comparison diagram that error sign ratio when aperture diameter changes changes with signal to noise ratio.As can be seen that With the increase of receiving aperture, the error sign ratio of two kinds of decision thresholds is decreased obviously, and communication performance improves.Fig. 4 is given at Optimal judgement thresholding and dynamic decision threshold schemes are used in the case of weak rapids, error sign ratio when shake variance changes is with signal to noise ratio The comparison diagram of change.As can be seen that with the reduction of shake variance, the error sign ratio of two kinds of decision thresholds is decreased obviously, Performance improves.Fig. 5,6 be given at it is strong rapid and it is weak it is rapid under the conditions of optimal judgement thresholding and dynamic decision threshold schemes, ripple is respectively adopted The comparison diagram that error sign ratio when beam width changes changes with signal to noise ratio.It can be seen that beam angle when low signal-to-noise ratio Narrower performance is better, and beam angle more wide feature is better when high s/n ratio.Because when signal to noise ratio is smaller, system Error sign ratio is mainly influenceed by transmit power, and beam angle is narrower, and the luminous power being collected into is bigger, the error sign ratio of system Can be better.And when signal to noise ratio increases to certain value, system error sign ratio is mainly influenceed by collimating fault.Now, it is wide The facula area that beam angle reaches receiving terminal is greater than narrow beam angle, that is, the light intensity received is big, then the mistake of system Symbol rate is lower.

The invention is not limited in above-described embodiment, on the basis of technical scheme disclosed by the invention, the skill of this area Art personnel are according to disclosed technology contents, it is not necessary to which performing creative labour can makes one to some of which technical characteristic A little to replace and deform, these are replaced and deformation is within the scope of the present invention.

Claims

1. the MPPM-FSO systems based on EW composite channel models demodulate error sign ratio computational methods firmly, comprise the steps：

1) it is distributed based on index weber EW, it is contemplated that the influence of point tolerance, establish the compound letter of point-to-point FSO Road model；

2) according to index weber EW turbulent flows and the cumulative distribution model F of the composite channel model of point tolerance_hAnd probability distribution mould (h) Type f_h(h) threshold value of optimal judgement thresholding and dynamic decision thresholding is determined；

3) mistaken verdict of each time slot under by index weber EW turbulent flows and point tolerance compound channel disturbed condition is calculated Probability；

2. the MPPM-FSO systems according to claim 1 based on EW composite channel models demodulate error sign ratio calculating side firmly Method, it is characterised in that in the step 1), be distributed based on index weber EW, it is contemplated that the influence of point tolerance, establish point-to-point FSO composite channel model, realized by following methods：

1a) establish point-to-point FSO composite channel model：

If x is sends signal sequence, y is reception signal sequence, then the mathematical modeling of channel is：

Y=Rhx+n

Wherein, n represent average be 0, variance beAdditive white Gaussian noise；R is photodetector responsiveness, and h refers to for obedience The channel fading of several webers of EW compound channels distributions, h=h_ah_p；Wherein, h_aTo obey the channel fading of index weber EW distributions, h_pFor point tolerance loss factor；

1b) according to step 1a) probability Distribution Models of obtained mathematical modeling structure index weber EW distributed modelsWith The probability Distribution Model of point tolerance

Wherein, h_aTo obey the channel fading of index weber EW distributions, h_pFor point tolerance loss factor；β is form parameter, β ＞ 0； α is in given observation space, collimates the average magnitude that the light beam of propagation is successfully received with the light beam of non-aligned propagation, α ＞ 0；η is Scale parameter, η ＞ 0；γ be equivalent beam angle and shake standard deviation ratio, γ=w_zeq/2σ_s；Wherein w_zeqFor equivalent ripple Beam width, σ_sTo shake standard deviation；A₀For the light that receiver receives when the offset between detector and beam center is 0 By force, A₀=[erf (v)]², wherein ν is the ratio of the beam angle at aperture and distance z,Wherein a is spy Survey the aperture radius of device, w_zFor the beam angle at the z of range transmission end；

1c) formula then can obtain the cumulative distribution model F of the composite channel model of atmospheric turbulance and point tolerance by above-mentioned 1., 2._h(h) With probability Distribution Model f_h(h)：

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In formula, Γ (a, x) is upper gamma incomplete function,For Mayer G-function；J is the counter of summation symbol.

3. the MPPM-FSO systems according to claim 2 based on EW composite channel models demodulate error sign ratio calculating side firmly Method, it is characterised in that in the step 2), the threshold settings of optimal judgement thresholding utilize Matlab by dichotomy to following formula Seek unknown number y_thObtain：

<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>M</mi> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <mi>&infin;</mi> </msubsup> <mi>exp</mi> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>2</mn> <msub> <mi>y</mi> <mrow> <mi>t</mi> <mi>h</mi> </mrow> </msub> <mi>g</mi> <mi>t</mi> </mrow> <mi>&mu;</mi> </mfrac> <mo>-</mo> <msup> <mi>gt</mi> <mn>2</mn> </msup> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mi>&alpha;&gamma;</mi> <mn>2</mn> </msup> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>A</mi> <mn>0</mn> </msub> <mi>&eta;</mi> </mrow> <mo>)</mo> </mrow> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> </msup> </mfrac> <msup> <mi>t</mi> <mrow> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mo>-</mo> <mn>1</mn> </mrow> </msup> <munderover> <mi>&Sigma;</mi> <mrow> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>&infin;</mi> </munderover> <mfrac> <mrow> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mi>&alpha;</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mrow> <mi>&alpha;</mi> <mo>-</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mi>j</mi> </msup> <mrow> <mi>j</mi> <mo>!</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mn>1</mn> <mo>-</mo> <mfrac> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mi>&beta;</mi> </mfrac> </mrow> </msup> </mrow> </mfrac> <mi>d</mi> <mi>t</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&times;</mo> <mi>&Gamma;</mi> <mrow> <mo>&lsqb;</mo> <mrow> <mn>1</mn> <mo>-</mo> <mfrac> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mi>&beta;</mi> </mfrac> <mo>,</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mfrac> <mi>t</mi> <mrow> <msub> <mi>A</mi> <mn>0</mn> </msub> <mi>&eta;</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mi>&beta;</mi> </msup> </mrow> <mo>&rsqb;</mo> </mrow> <mi>d</mi> <mi>t</mi> <mrow> <mo>&lsqb;</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>e</mi> <mi>r</mi> <mi>f</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>y</mi> <mrow> <mi>t</mi> <mi>h</mi> </mrow> </msub> <msqrt> <mi>g</mi> </msqrt> </mrow> <mi>&mu;</mi> </mfrac> <mo>)</mo> </mrow> </mrow> <mo>&rsqb;</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>N</mi> <mo>-</mo> <mi>M</mi> </mrow> <mo>)</mo> </mrow> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <mi>&infin;</mi> </msubsup> <mfrac> <mrow> <msup> <mi>&alpha;&gamma;</mi> <mn>2</mn> </msup> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>A</mi> <mn>0</mn> </msub> <mi>&eta;</mi> </mrow> <mo>)</mo> </mrow> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> </msup> </mfrac> <msup> <mi>t</mi> <mrow> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mo>-</mo> <mn>1</mn> </mrow> </msup> <munderover> <mi>&Sigma;</mi> <mrow> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>&infin;</mi> </munderover> <mfrac> <mrow> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mi>&alpha;</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mrow> <mi>&alpha;</mi> <mo>-</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mi>j</mi> </msup> <mrow> <mi>j</mi> <mo>!</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mn>1</mn> <mo>-</mo> <mfrac> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mi>&beta;</mi> </mfrac> </mrow> </msup> </mrow> </mfrac> <mi>&Gamma;</mi> <mrow> <mo>&lsqb;</mo> <mrow> <mn>1</mn> <mo>-</mo> <mfrac> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mi>&beta;</mi> </mfrac> <mo>,</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mfrac> <mi>t</mi> <mrow> <msub> <mi>A</mi> <mn>0</mn> </msub> <mi>&eta;</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mi>&beta;</mi> </msup> </mrow> <mo>&rsqb;</mo> </mrow> <mi>d</mi> <mi>t</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&times;</mo> <mi>e</mi> <mi>r</mi> <mi>f</mi> <mi>c</mi> <mrow> <mo>&lsqb;</mo> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>y</mi> <mrow> <mi>t</mi> <mi>h</mi> </mrow> </msub> <mo>-</mo> <mi>&mu;</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <msqrt> <mi>g</mi> </msqrt> </mrow> <mi>&mu;</mi> </mfrac> <mo>&rsqb;</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced>

In formula, number of time slots that a block of information that N is modulated by MPPM includes, M is the pulse in this N number of number of time slots Number；T is the integration variable of integrand；y_thFor threshold value；μ is coefficient, and μ²=(RNP_t/M)², R is photoelectric transformation efficiency, P_t For the average emitted power of transmitting terminal；G is signal to noise ratio；Exp () is the exponential function using natural number e the bottom of as；Γ () is gamma Function；Erf () is error function；Erfc () is cosine error function；J is the counter of summation symbol.

4. the MPPM-FSO systems according to claim 1 based on EW composite channel models demodulate error sign ratio calculating side firmly Method, it is characterised in that the threshold value of dynamic decision thresholding is to detect the signal of each time slot, takes the 200ns of its signal leading edge, Signal intensity according to being detected in this period takes the value of setting dynamic decision thresholding.

5. the MPPM-FSO systems according to claim 3 based on EW composite channel models demodulate error sign ratio calculating side firmly Method, it is characterised in that in the step 3), calculate each time slot in the compound channel by index weber EW turbulent flows and point tolerance The probability of mistaken verdict under disturbed condition, realized by following methods：

When 3a) using optimal judgement thresholding, it is known that the probability Distribution Model f of compound channel_h(h) mathematical model and in step 1), Can then calculate by non-slot signal f under index weber EW turbulent flows and point tolerance disturbed condition (y | y₀) and time slot signal f (y | y₁) probability density function of sample value is respectively：

<mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>y</mi> <mo>|</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <msqrt> <mrow> <mn>2</mn> <mi>&pi;</mi> </mrow> </msqrt> <msub> <mi>&sigma;</mi> <mi>n</mi> </msub> </mrow> </mfrac> <mi>exp</mi> <mrow> <mo>(</mo> <mo>-</mo> <mfrac> <msup> <mi>y</mi> <mn>2</mn> </msup> <mrow> <mn>2</mn> <msup> <msub> <mi>&sigma;</mi> <mi>n</mi> </msub> <mn>2</mn> </msup> </mrow> </mfrac> <mo>)</mo> </mrow> </mrow>

<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>y</mi> <mo>|</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mi>&mu;</mi> </mfrac> <msub> <mi>f</mi> <mi>h</mi> </msub> <mrow> <mo>(</mo> <mfrac> <mi>y</mi> <mi>&mu;</mi> </mfrac> <mo>)</mo> </mrow> <mo>*</mo> <msub> <mi>f</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mi>&mu;</mi> </mfrac> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <mi>&infin;</mi> </msubsup> <mfrac> <mrow> <msup> <mi>&alpha;&gamma;</mi> <mn>2</mn> </msup> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>&eta;A</mi> <mn>0</mn> </msub> </mrow> <mo>)</mo> </mrow> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> </msup> </mfrac> <msup> <mrow> <mo>(</mo> <mfrac> <mi>h</mi> <mi>&mu;</mi> </mfrac> <mo>)</mo> </mrow> <mrow> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mo>-</mo> <mn>1</mn> </mrow> </msup> <munderover> <mi>&Sigma;</mi> <mrow> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>&infin;</mi> </munderover> <mfrac> <mrow> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mi>&alpha;</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mrow> <mi>&alpha;</mi> <mo>-</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mi>j</mi> </msup> <mrow> <mi>j</mi> <mo>!</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mn>1</mn> <mo>-</mo> <mfrac> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mi>&beta;</mi> </mfrac> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>&Gamma;</mi> <mrow> <mo>&lsqb;</mo> <mrow> <mn>1</mn> <mo>-</mo> <mfrac> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mi>&beta;</mi> </mfrac> <mo>,</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mfrac> <msub> <mi>h</mi> <mi>a</mi> </msub> <mrow> <msub> <mi>A</mi> <mn>0</mn> </msub> <mi>&eta;</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mi>&beta;</mi> </msup> </mrow> <mo>&rsqb;</mo> </mrow> <mo>&times;</mo> <mfrac> <mn>1</mn> <mrow> <msqrt> <mrow> <mn>2</mn> <mi>&pi;</mi> </mrow> </msqrt> <msub> <mi>&sigma;</mi> <mi>n</mi> </msub> </mrow> </mfrac> <mi>exp</mi> <mrow> <mo>(</mo> <mrow> <mo>-</mo> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mi>y</mi> <mo>-</mo> <mi>h</mi> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mrow> <mn>2</mn> <msup> <msub> <mi>&sigma;</mi> <mi>n</mi> </msub> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>h</mi> </mrow> </mtd> </mtr> </mtable> </mfenced>

In formula, σ_n ²For the variance of additive white Gaussian noise；f(y|y₁) it is to have the probability density function of signal slot sample value, f (y |y₀) be no signal time slot sample value probability density function；

By non-slot signal f (y | y₀) mistaken verdict probability, i.e. false-alarm probability P_fFor：

<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>P</mi> <mi>f</mi> </msub> <mo>=</mo> <msubsup> <mo>&Integral;</mo> <msub> <mi>y</mi> <mrow> <mi>t</mi> <mi>h</mi> </mrow> </msub> <mi>&infin;</mi> </msubsup> <mi>f</mi> <mrow> <mo>(</mo> <mi>y</mi> <mo>|</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mi>d</mi> <mi>y</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mn>1</mn> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>&lsqb;</mo> <mn>1</mn> <mo>+</mo> <mi>e</mi> <mi>r</mi> <mi>f</mi> <mrow> <mo>(</mo> <mfrac> <msub> <mi>y</mi> <mrow> <mi>t</mi> <mi>h</mi> </mrow> </msub> <mrow> <msqrt> <mn>2</mn> </msqrt> <msub> <mi>&sigma;</mi> <mi>n</mi> </msub> </mrow> </mfrac> <mo>)</mo> </mrow> <mo>&rsqb;</mo> </mrow> </mtd> </mtr> </mtable> </mfenced>

By time slot signal f (y | y₁) mistaken verdict probability, i.e. false dismissal probability P_mFor：

<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>P</mi> <mi>m</mi> </msub> <mo>=</mo> <msubsup> <mo>&Integral;</mo> <mrow> <mo>-</mo> <mi>&infin;</mi> </mrow> <msub> <mi>y</mi> <mrow> <mi>t</mi> <mi>h</mi> </mrow> </msub> </msubsup> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>y</mi> <mo>|</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>y</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mn>1</mn> <mo>-</mo> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <mi>&infin;</mi> </msubsup> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>e</mi> <mi>r</mi> <mi>f</mi> <mi>c</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>y</mi> <mrow> <mi>t</mi> <mi>h</mi> </mrow> </msub> <mo>-</mo> <mi>&mu;</mi> <mi>t</mi> </mrow> <mrow> <msqrt> <mn>2</mn> </msqrt> <msub> <mi>&sigma;</mi> <mi>n</mi> </msub> </mrow> </mfrac> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mi>&alpha;&gamma;</mi> <mn>2</mn> </msup> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>A</mi> <mn>0</mn> </msub> <mi>&eta;</mi> </mrow> <mo>)</mo> </mrow> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> </msup> </mfrac> <msup> <mi>t</mi> <mrow> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mo>-</mo> <mn>1</mn> </mrow> </msup> <munderover> <mi>&Sigma;</mi> <mrow> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>&infin;</mi> </munderover> <mfrac> <mrow> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mi>&alpha;</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mrow> <mi>&alpha;</mi> <mo>-</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mi>j</mi> </msup> <mrow> <mi>j</mi> <mo>!</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mn>1</mn> <mo>-</mo> <mfrac> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mi>&beta;</mi> </mfrac> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&times;</mo> <mi>&Gamma;</mi> <mrow> <mo>&lsqb;</mo> <mrow> <mn>1</mn> <mo>-</mo> <mfrac> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mi>&beta;</mi> </mfrac> <mo>,</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mfrac> <mi>t</mi> <mrow> <msub> <mi>A</mi> <mn>0</mn> </msub> <mi>&eta;</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mi>&beta;</mi> </msup> </mrow> <mo>&rsqb;</mo> </mrow> <mi>d</mi> <mi>t</mi> </mrow> </mtd> </mtr> </mtable> </mfenced>

When 3b) using dynamic decision thresholding, it is known that the probability that (N, M) MPPM block of informations empty slots occur is N-M/N, during non-NULL The probability that gap occurs is M/N, then probability of miscarriage of justice of each time slot when existing for only white Gaussian noise is

Mathematical model in the probability Distribution Model and step 1) of known compound channel, then each time slot can be calculated in index weber EW turbulent flows and the probability of the mistaken verdict under point tolerance disturbed condition：

<mrow> <mtable> <mtr> <mtd> <mrow> <mi>S</mi> <mi>L</mi> <mi>E</mi> <mi>R</mi> <mo>=</mo> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <mi>&infin;</mi> </msubsup> <mi>S</mi> <mi>L</mi> <mi>E</mi> <mi>R</mi> <mrow> <mo>(</mo> <mrow> <mi>g</mi> <mo>,</mo> <mi>h</mi> </mrow> <mo>)</mo> </mrow> <msub> <mi>f</mi> <mi>h</mi> </msub> <mrow> <mo>(</mo> <mi>h</mi> <mo>)</mo> </mrow> <mi>d</mi> <mi>h</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <mi>&infin;</mi> </msubsup> <mfrac> <mi>M</mi> <mrow> <mn>2</mn> <mi>N</mi> </mrow> </mfrac> <mi>e</mi> <mi>r</mi> <mi>f</mi> <mi>c</mi> <mrow> <mo>&lsqb;</mo> <mrow> <msqrt> <mrow> <msup> <mi>gh</mi> <mn>2</mn> </msup> </mrow> </msqrt> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>-</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msup> <mi>gh</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mi>ln</mi> <mfrac> <mrow> <mi>N</mi> <mo>-</mo> <mi>M</mi> </mrow> <mi>M</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>&rsqb;</mo> </mrow> <mfrac> <mrow> <msup> <mi>&alpha;&gamma;</mi> <mn>2</mn> </msup> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>A</mi> <mn>0</mn> </msub> <mi>&eta;</mi> </mrow> <mo>)</mo> </mrow> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> </msup> </mfrac> <msup> <mi>t</mi> <mrow> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mo>-</mo> <mn>1</mn> </mrow> </msup> <munderover> <mi>&Sigma;</mi> <mrow> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>&infin;</mi> </munderover> <mfrac> <mrow> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mi>&alpha;</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mrow> <mi>&alpha;</mi> <mo>-</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&times;</mo> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mi>j</mi> </msup> <mrow> <mi>j</mi> <mo>!</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mn>1</mn> <mo>-</mo> <mfrac> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mi>&beta;</mi> </mfrac> </mrow> </msup> </mrow> </mfrac> <mi>&Gamma;</mi> <mrow> <mo>&lsqb;</mo> <mrow> <mn>1</mn> <mo>-</mo> <mfrac> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mi>&beta;</mi> </mfrac> <mo>,</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mfrac> <mi>t</mi> <mrow> <msub> <mi>A</mi> <mn>0</mn> </msub> <mi>&eta;</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mi>&beta;</mi> </msup> </mrow> <mo>&rsqb;</mo> </mrow> <mi>d</mi> <mi>h</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <mi>&infin;</mi> </msubsup> <mfrac> <mrow> <mi>N</mi> <mo>-</mo> <mi>M</mi> </mrow> <mrow> <mn>2</mn> <mi>N</mi> </mrow> </mfrac> <mi>e</mi> <mi>r</mi> <mi>f</mi> <mi>c</mi> <mrow> <mo>&lsqb;</mo> <mrow> <msqrt> <mrow> <msup> <mi>gh</mi> <mn>2</mn> </msup> </mrow> </msqrt> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msup> <mi>gh</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mi>ln</mi> <mfrac> <mrow> <mi>N</mi> <mo>-</mo> <mi>M</mi> </mrow> <mi>M</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>&rsqb;</mo> </mrow> <mfrac> <mrow> <msup> <mi>&alpha;&gamma;</mi> <mn>2</mn> </msup> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>A</mi> <mn>0</mn> </msub> <mi>&eta;</mi> </mrow> <mo>)</mo> </mrow> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> </msup> </mfrac> <msup> <mi>t</mi> <mrow> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mo>-</mo> <mn>1</mn> </mrow> </msup> <munderover> <mi>&Sigma;</mi> <mrow> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>&infin;</mi> </munderover> <mfrac> <mrow> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mi>&alpha;</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mrow> <mi>&alpha;</mi> <mo>-</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&times;</mo> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mi>j</mi> </msup> <mrow> <mi>j</mi> <mo>!</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mn>1</mn> <mo>-</mo> <mfrac> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mi>&beta;</mi> </mfrac> </mrow> </msup> </mrow> </mfrac> <mi>&Gamma;</mi> <mrow> <mo>&lsqb;</mo> <mrow> <mn>1</mn> <mo>-</mo> <mfrac> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mi>&beta;</mi> </mfrac> <mo>,</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mfrac> <mi>t</mi> <mrow> <msub> <mi>A</mi> <mn>0</mn> </msub> <mi>&eta;</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mi>&beta;</mi> </msup> </mrow> <mo>&rsqb;</mo> </mrow> <mi>d</mi> <mi>h</mi> </mrow> </mtd> </mtr> </mtable> <mo>.</mo> </mrow>

6. the MPPM-FSO systems based on EW composite channel models demodulate error sign ratio calculating side firmly according to claim 5 Method, it is characterised in that in the step 4), using Gauss Laguerre polynomials, obtain the closure expression formula of error sign ratio, be logical Following methods are crossed to realize：

When 4a) using optimal judgement thresholding, for (N, M) MPPM, the probability correctly demodulated is P_c=(1-P_m)^M(1-P_f)^N-M, its The middle correct probability of pulse slot and non-pulse time slot is

<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <mn>1</mn> <mo>-</mo> <msub> <mi>P</mi> <mi>m</mi> </msub> <mo>=</mo> <msubsup> <mo>&Integral;</mo> <msub> <mi>y</mi> <mrow> <mi>t</mi> <mi>h</mi> </mrow> </msub> <mi>&infin;</mi> </msubsup> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>y</mi> <mo>|</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>y</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <mi>&infin;</mi> </msubsup> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>e</mi> <mi>r</mi> <mi>f</mi> <mi>c</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>y</mi> <mrow> <mi>t</mi> <mi>h</mi> </mrow> </msub> <mo>-</mo> <mi>&mu;</mi> <mi>t</mi> </mrow> <mrow> <msqrt> <mn>2</mn> </msqrt> <msub> <mi>&sigma;</mi> <mi>n</mi> </msub> </mrow> </mfrac> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mi>&alpha;&gamma;</mi> <mn>2</mn> </msup> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>A</mi> <mn>0</mn> </msub> <mi>&eta;</mi> </mrow> <mo>)</mo> </mrow> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> </msup> </mfrac> <msup> <mi>t</mi> <mrow> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mo>-</mo> <mn>1</mn> </mrow> </msup> <munderover> <mi>&Sigma;</mi> <mrow> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>&infin;</mi> </munderover> <mfrac> <mrow> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mi>&alpha;</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mrow> <mi>&alpha;</mi> <mo>-</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&times;</mo> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mi>j</mi> </msup> <mrow> <mi>j</mi> <mo>!</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mn>1</mn> <mo>-</mo> <mfrac> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mi>&beta;</mi> </mfrac> </mrow> </msup> </mrow> </mfrac> <mi>&Gamma;</mi> <mrow> <mo>&lsqb;</mo> <mrow> <mn>1</mn> <mo>-</mo> <mfrac> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mi>&beta;</mi> </mfrac> <mo>,</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mfrac> <mi>t</mi> <mrow> <msub> <mi>A</mi> <mn>0</mn> </msub> <mi>&eta;</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mi>&beta;</mi> </msup> </mrow> <mo>&rsqb;</mo> </mrow> <mi>d</mi> <mi>t</mi> </mrow> </mtd> </mtr> </mtable> </mfenced>

<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <mn>1</mn> <mo>-</mo> <msub> <mi>P</mi> <mi>f</mi> </msub> <mo>=</mo> <msubsup> <mo>&Integral;</mo> <mrow> <mo>-</mo> <mi>&infin;</mi> </mrow> <msub> <mi>y</mi> <mrow> <mi>t</mi> <mi>h</mi> </mrow> </msub> </msubsup> <mfrac> <mn>1</mn> <mrow> <msqrt> <mrow> <mn>2</mn> <mi>&pi;</mi> </mrow> </msqrt> <msub> <mi>&sigma;</mi> <mi>n</mi> </msub> </mrow> </mfrac> <mi>exp</mi> <mrow> <mo>(</mo> <mrow> <mo>-</mo> <mfrac> <msup> <mi>y</mi> <mn>2</mn> </msup> <mrow> <mn>2</mn> <msup> <msub> <mi>&sigma;</mi> <mi>n</mi> </msub> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>y</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mn>1</mn> <mo>-</mo> <msubsup> <mo>&Integral;</mo> <msub> <mi>y</mi> <mrow> <mi>t</mi> <mi>h</mi> </mrow> </msub> <mi>&infin;</mi> </msubsup> <mfrac> <mn>1</mn> <mrow> <msqrt> <mrow> <mn>2</mn> <mi>&pi;</mi> </mrow> </msqrt> <msub> <mi>&sigma;</mi> <mi>n</mi> </msub> </mrow> </mfrac> <mi>exp</mi> <mrow> <mo>(</mo> <mrow> <mo>-</mo> <mfrac> <msup> <mi>y</mi> <mn>2</mn> </msup> <mrow> <mn>2</mn> <msup> <msub> <mi>&sigma;</mi> <mi>n</mi> </msub> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>y</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>&lsqb;</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>e</mi> <mi>r</mi> <mi>f</mi> <mrow> <mo>(</mo> <mfrac> <msub> <mi>y</mi> <mrow> <mi>t</mi> <mi>h</mi> </mrow> </msub> <mrow> <msqrt> <mn>2</mn> </msqrt> <msub> <mi>&sigma;</mi> <mi>n</mi> </msub> </mrow> </mfrac> <mo>)</mo> </mrow> </mrow> <mo>&rsqb;</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced>

Then error sign ratio is

<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>P</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>P</mi> <mi>c</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mn>1</mn> <mo>-</mo> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mi>N</mi> </msup> </mfrac> <msup> <mrow> <mo>&lsqb;</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>e</mi> <mi>r</mi> <mi>f</mi> <mrow> <mo>(</mo> <mfrac> <msub> <mi>y</mi> <mrow> <mi>t</mi> <mi>h</mi> </mrow> </msub> <mrow> <msqrt> <mn>2</mn> </msqrt> <msub> <mi>&sigma;</mi> <mi>n</mi> </msub> </mrow> </mfrac> <mo>)</mo> </mrow> </mrow> <mo>&rsqb;</mo> </mrow> <mrow> <mi>N</mi> <mo>-</mo> <mi>M</mi> </mrow> </msup> <mo>{</mo> <mn>2</mn> <mo>-</mo> <mfrac> <mn>1</mn> <msqrt> <mi>&pi;</mi> </msqrt> </mfrac> <mi>exp</mi> <mrow> <mo>(</mo> <mrow> <mo>-</mo> <mfrac> <mrow> <msup> <msub> <mi>gy</mi> <mrow> <mi>t</mi> <mi>h</mi> </mrow> </msub> <mn>2</mn> </msup> </mrow> <msup> <mi>&mu;</mi> <mn>2</mn> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <msup> <mi>&alpha;&gamma;</mi> <mn>2</mn> </msup> </mrow> <mi>&beta;</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&times;</mo> <munderover> <mi>&Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msub> <mi>H</mi> <mi>i</mi> </msub> <mi>exp</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>y</mi> <mrow> <mi>t</mi> <mi>h</mi> </mrow> </msub> <msqrt> <mrow> <msub> <mi>x</mi> <mi>i</mi> </msub> <mi>g</mi> </mrow> </msqrt> </mrow> <mi>&mu;</mi> </mfrac> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mfrac> <msqrt> <msub> <mi>x</mi> <mi>i</mi> </msub> </msqrt> <mrow> <msub> <mi>&eta;A</mi> <mn>0</mn> </msub> <msqrt> <mi>g</mi> </msqrt> </mrow> </mfrac> <mo>)</mo> </mrow> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> </msup> <munderover> <mi>&Sigma;</mi> <mrow> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>&infin;</mi> </munderover> <mfrac> <mrow> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mi>&alpha;</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mrow> <mi>&alpha;</mi> <mo>-</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mi>j</mi> </msup> <mrow> <mi>j</mi> <mo>!</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mn>1</mn> <mo>-</mo> <mfrac> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mi>&beta;</mi> </mfrac> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&times;</mo> <msubsup> <mi>G</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> <mrow> <mn>2</mn> <mo>,</mo> <mn>0</mn> </mrow> </msubsup> <mrow> <mo>&lsqb;</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mfrac> <msqrt> <msub> <mi>x</mi> <mi>i</mi> </msub> </msqrt> <mrow> <msub> <mi>&eta;A</mi> <mn>0</mn> </msub> <msqrt> <mi>g</mi> </msqrt> </mrow> </mfrac> <mo>)</mo> </mrow> <mi>&beta;</mi> </msup> <mo>|</mo> <mtable> <mtr> <mtd> <mrow> <mn>1</mn> <mo>-</mo> <mfrac> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mi>&beta;</mi> </mfrac> <mo>,</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>-</mo> <mfrac> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mi>&beta;</mi> </mfrac> <mo>,</mo> <mo>-</mo> <mfrac> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mi>&beta;</mi> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>&rsqb;</mo> </mrow> <msup> <mo>}</mo> <mi>M</mi> </msup> </mrow> </mtd> </mtr> </mtable> </mfenced>

In formula, H_iIt is the weight in generalized Laguerre formula；x_iFor i-th of Generalized Gaussian Laguerre polynomials；K is broad sense The total item of Gauss Laguerre polynomials；

When 4b) using dynamic decision thresholding, for (N, M) MPPM, the probability correctly demodulated is P_c=(1-SLER)^N, then erratum number Rate is

<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>P</mi> <mi>s</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>P</mi> <mi>c</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mn>1</mn> <mo>-</mo> <mfrac> <mi>M</mi> <mrow> <mn>2</mn> <mi>N</mi> <msqrt> <mi>&pi;</mi> </msqrt> </mrow> </mfrac> <munderover> <mi>&Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msub> <mi>H</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <msub> <mi>x</mi> <mi>i</mi> </msub> </mrow> </mfrac> <mi>ln</mi> <mfrac> <mrow> <mi>N</mi> <mo>-</mo> <mi>M</mi> </mrow> <mi>M</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mi>N</mi> <mo>-</mo> <mi>M</mi> </mrow> <mi>M</mi> </mfrac> <mo>)</mo> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>-</mo> <mfrac> <mn>1</mn> <mrow> <mn>16</mn> <msub> <mi>x</mi> <mi>i</mi> </msub> </mrow> </mfrac> <mi>ln</mi> <mfrac> <mrow> <mi>N</mi> <mo>-</mo> <mi>M</mi> </mrow> <mi>M</mi> </mfrac> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mn>2</mn> <msqrt> <msub> <mi>x</mi> <mi>i</mi> </msub> </msqrt> </mrow> <mrow> <msub> <mi>&eta;A</mi> <mn>0</mn> </msub> <msqrt> <mi>g</mi> </msqrt> </mrow> </mfrac> <mo>)</mo> </mrow> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&times;</mo> <mfrac> <mrow> <msup> <mi>&alpha;&gamma;</mi> <mn>2</mn> </msup> </mrow> <mi>&beta;</mi> </mfrac> <munderover> <mi>&Sigma;</mi> <mrow> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>&infin;</mi> </munderover> <mfrac> <mrow> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mi>&alpha;</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mrow> <mi>&alpha;</mi> <mo>-</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mi>j</mi> </msup> <mrow> <mi>j</mi> <mo>!</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mn>1</mn> <mo>-</mo> <mfrac> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mi>&beta;</mi> </mfrac> </mrow> </msup> </mrow> </mfrac> <msubsup> <mi>G</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> <mrow> <mn>2</mn> <mo>,</mo> <mn>0</mn> </mrow> </msubsup> <mrow> <mo>&lsqb;</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mn>2</mn> <msqrt> <msub> <mi>x</mi> <mi>i</mi> </msub> </msqrt> </mrow> <mrow> <msub> <mi>&eta;A</mi> <mn>0</mn> </msub> <msqrt> <mi>g</mi> </msqrt> </mrow> </mfrac> <mo>)</mo> </mrow> <mi>&beta;</mi> </msup> <mo>|</mo> <mtable> <mtr> <mtd> <mrow> <mn>1</mn> <mo>-</mo> <mfrac> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mi>&beta;</mi> </mfrac> <mo>,</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>-</mo> <mfrac> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mi>&beta;</mi> </mfrac> <mo>,</mo> <mo>-</mo> <mfrac> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mi>&beta;</mi> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>&rsqb;</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mfrac> <mrow> <mi>N</mi> <mo>-</mo> <mi>M</mi> </mrow> <mrow> <mn>2</mn> <mi>N</mi> <msqrt> <mi>&pi;</mi> </msqrt> </mrow> </mfrac> <munderover> <mi>&Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msub> <mi>H</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>-</mo> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <msub> <mi>x</mi> <mi>i</mi> </msub> </mrow> </mfrac> <mi>ln</mi> <mfrac> <mrow> <mi>N</mi> <mo>-</mo> <mi>M</mi> </mrow> <mi>M</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mi>N</mi> <mo>-</mo> <mi>M</mi> </mrow> <mi>M</mi> </mfrac> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>-</mo> <mfrac> <mn>1</mn> <mrow> <mn>16</mn> <msub> <mi>x</mi> <mi>i</mi> </msub> </mrow> </mfrac> <mi>ln</mi> <mfrac> <mrow> <mi>N</mi> <mo>-</mo> <mi>M</mi> </mrow> <mi>M</mi> </mfrac> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mn>2</mn> <msqrt> <msub> <mi>x</mi> <mi>i</mi> </msub> </msqrt> </mrow> <mrow> <msub> <mi>&eta;A</mi> <mn>0</mn> </msub> <msqrt> <mi>g</mi> </msqrt> </mrow> </mfrac> <mo>)</mo> </mrow> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&times;</mo> <mfrac> <mrow> <msup> <mi>&alpha;&gamma;</mi> <mn>2</mn> </msup> </mrow> <mi>&beta;</mi> </mfrac> <munderover> <mi>&Sigma;</mi> <mrow> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>&infin;</mi> </munderover> <mfrac> <mrow> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mi>&alpha;</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>&Gamma;</mi> <mrow> <mo>(</mo> <mrow> <mi>&alpha;</mi> <mo>-</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mi>j</mi> </msup> <mrow> <mi>j</mi> <mo>!</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mn>1</mn> <mo>-</mo> <mfrac> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mi>&beta;</mi> </mfrac> </mrow> </msup> </mrow> </mfrac> <msubsup> <mi>G</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> <mrow> <mn>2</mn> <mo>,</mo> <mn>0</mn> </mrow> </msubsup> <mrow> <mo>&lsqb;</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>j</mi> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mn>2</mn> <msqrt> <msub> <mi>x</mi> <mi>i</mi> </msub> </msqrt> </mrow> <mrow> <msub> <mi>&eta;A</mi> <mn>0</mn> </msub> <msqrt> <mi>g</mi> </msqrt> </mrow> </mfrac> <mo>)</mo> </mrow> <mi>&beta;</mi> </msup> <mo>|</mo> <mtable> <mtr> <mtd> <mrow> <mn>1</mn> <mo>-</mo> <mfrac> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mi>&beta;</mi> </mfrac> <mo>,</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>-</mo> <mfrac> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mi>&beta;</mi> </mfrac> <mo>,</mo> <mo>-</mo> <mfrac> <msup> <mi>&gamma;</mi> <mn>2</mn> </msup> <mi>&beta;</mi> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>&rsqb;</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>.</mo> </mrow>